let cn be Real; :: thesis: for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let K0, B0 be Subset of (TOP-REAL 2); :: thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); :: thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous )
set sn = sqrt (1 - (cn ^2));
set p0 = |[cn,(- (sqrt (1 - (cn ^2))))]|;
assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; :: thesis: f is continuous
then cn ^2 < 1 ^2 by SQUARE_1:50;
then 1 - (cn ^2) > 0 by XREAL_1:50;
then ( |[cn,(- (sqrt (1 - (cn ^2))))]| `2 = - (sqrt (1 - (cn ^2))) & - (- (sqrt (1 - (cn ^2)))) > 0 ) by EUCLID:52, SQUARE_1:25;
then A2: |[cn,(- (sqrt (1 - (cn ^2))))]| `2 < 0 ;
then |[cn,(- (sqrt (1 - (cn ^2))))]| in K0 by A1, JGRAPH_2:3;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
not |[cn,(- (sqrt (1 - (cn ^2))))]| in {(0. (TOP-REAL 2))} by A2, JGRAPH_2:3, TARSKI:def 1;
then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A1, XBOOLE_0:def 5;
A3: K1 c= D

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in K1 or x in D )
assume x in K1 ; :: thesis: x in D
then consider p2 being Point of (TOP-REAL 2) such that
A4: p2 = x and
p2 `2 <= 0 and
A5: p2 <> 0. (TOP-REAL 2) by A1;
not p2 in {(0. (TOP-REAL 2))} by A5, TARSKI:def 1;
hence x in D by A1, A4, XBOOLE_0:def 5; :: thesis: verum

end;

for p being Point of ((TOP-REAL 2) | K1)
for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V )

proof

let p be Point of ((TOP-REAL 2) | K1); :: thesis: for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V )
let V be Subset of ((TOP-REAL 2) | D); :: thesis: ( f . p in V & V is open implies ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V ) )
assume that
A6: f . p in V and
A7: V is open ; :: thesis: ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V )
consider V2 being Subset of (TOP-REAL 2) such that
A8: V2 is open and
A9: V2 /\ ([#] ((TOP-REAL 2) | D)) = V by A7, TOPS_2:24;
reconsider W2 = V2 /\ ([#] ((TOP-REAL 2) | K1)) as Subset of ((TOP-REAL 2) | K1) ;
A10: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def 5;
then A11: f . p = (cn -FanMorphN) . p by A1, FUNCT_1:49;
A12: f .: W2 c= V

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in f .: W2 or y in V )
assume y in f .: W2 ; :: thesis: y in V
then consider x being object such that
A13: x in dom f and
A14: x in W2 and
A15: y = f . x by FUNCT_1:def 6;
f is Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) ;
then dom f = K1 by A10, FUNCT_2:def 1;
then consider p4 being Point of (TOP-REAL 2) such that
A16: x = p4 and
A17: p4 `2 <= 0 and
p4 <> 0. (TOP-REAL 2) by A1, A13;
A18: p4 in V2 by A14, A16, XBOOLE_0:def 4;
p4 in [#] ((TOP-REAL 2) | K1) by A13, A16;
then p4 in D by A3, A10;
then A19: p4 in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def 5;
f . p4 = (cn -FanMorphN) . p4 by A1, A10, A13, A16, FUNCT_1:49
.= p4 by A17, Th49 ;
hence y in V by A9, A15, A16, A18, A19, XBOOLE_0:def 4; :: thesis: verum

end;

p in the carrier of ((TOP-REAL 2) | K1) ;
then consider q being Point of (TOP-REAL 2) such that
A20: q = p and
A21: q `2 <= 0 and
q <> 0. (TOP-REAL 2) by A1, A10;
(cn -FanMorphN) . q = q by A21, Th49;
then p in V2 by A6, A9, A11, A20, XBOOLE_0:def 4;
then A22: p in W2 by XBOOLE_0:def 4;
W2 is open by A8, TOPS_2:24;
hence ex W being Subset of ((TOP-REAL 2) | K1) st
( p in W & W is open & f .: W c= V ) by A22, A12; :: thesis: verum

end;

hence f is continuous by JGRAPH_2:10; :: thesis: verum